Abstract
Quaternions are an extension of the idea of complex numbers to four dimensions. We discuss the iteration of linear and quadratic functions of the quaternions, and examine the rôle played by regularity (the analog of complex analyticity), in this context. In contrast to the complex case, regularity is not automatically preserved by iteration of quaternion functions. We find that demanding preservation of regularity is too restrictive, yielding very little new beyond the complex case. The quaternion generalisation of the Mandelbrot set is described.
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